The introduction to StoNEDΒΆ

Combing virtues of SFA and DEA in a unified framework, Stochastic Nonparametric Envelopment of Data (StoNED) (kuosmanen, 2006) uses a composed error term to model both inefficiency \(u\) and noise \(v\) without assuming a functional form of \(f\). Analogous to the COLS/C \(^2\) NLS estimators, the StoNED estimator consists of the following four steps:

  • Step 1: Estimating the conditional mean \(E[y_i \,| \, \boldsymbol{x}_i]\) using CNLS estimator

  • Step 2: Estimating the expected inefficiency \(\mu\) based on the residual \(\varepsilon_i^{CNLS}\)

  • Step 3: Estimating the StoNED frontier \(\hat{f}^{StoNED}\) based on the \(\hat{\mu}\)

  • Step 4: Estimating firm-specific inefficiencies \(E[u_i \mid \varepsilon_i^{CNLS}]\)

Beside the CNLS estimator, we can apply other convex regression approaches such as ICNLS and CNLS-DDF to estimate the conditional mean in the first step (see Keshvari and Kuosmanen, 2013; Kuosmanen and Johnson, 2017). However, the quantile and expectile related estimators introduced in Examples can not be integrated into StoNED framework at present.

After obtaining the residuals (e.g., \(\hat{\varepsilon}_i^{CNLS}\)) from the convex regression approaches, one can estimate the expected value of the inefficiency term \(\mu = E(u_i)\). In practice, three commonly used methods are available to estimate the expected inefficiency \(\mu\): method of moments (Aigner et al., 1977), quasi-likelihood estimation (Fan et al., 1996), and the kernel deconvolution estimation (Hall and Simar, 2002). We will next briefly review these three approaches and focus on demonstrating the application of pyStoNED; see more detailed theoretical introduction in Kuosmanen et al. (2015).